RATIONAL RECIPROCITY LAWS

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1 RATIONAL RECIPROCITY LAWS MARK BUDDEN 1 10/7/05 The urose of this note is to rovide an overview of Rational Recirocity and in articular, of Scholz s recirocity law for the non-number theorist. In the first art, we will describe the background in number theory that will be necessary for a comlete understanding of the material to be discussed in the second art. The second art focuses on a roof of Scholz s recirocity law using the slitting of minimal olynomials and considers ways in which this law can be extended. 1. Algebraic Number Fields Here, we focus on the necessary asects of number theory that will be used throughout the remainder of this note. By an algebraic number field K, we mean a finite dimensional extension of Q. If K and L are two algebraic number fields satisfying K L, then L can be viewed as a K-vector sace and we denote its dimension by [L : K]. By the Primitive Element Theorem, there exists a L such that L Ka. The element a is algebraic over K it is the root of a nonzero olynomial in K[x] - see Theorem.1 of Jacobson [J]. There exists a uniue monic irreducible olynomial in K[x] with a as a root. This olynomial is referred to as the minimal olynomial of a over K and its degree is eual to the dimension of L over K. The Galois grou GalL/K is defined to be the grou of all automorhisms of L that fix K. Now we define an imortant subring of an algebraic number field. Such a subring will mimic the role that Z lays as a subring of Q. An element b K is called an algebraic integer if it is the root of a monic olynomial in Z[x]. The set of all algebraic integers in K form a ring, called the ring of integers in K, that is denoted by O K. If L/K is an extension of algebraic number fields, then O L K O K Quadratic Fields. Consider the uadratic field Q d where d 0 is a suare-free integer. It is a simle exercise see Proosition of [IR] to show that [ 1+ Z ] d O Q d if d 1mod Z[ d] if d, 3mod. Furthermore, the units of Q d are given by ±1} if d < 3 or d O Q ±1, ±i} if d 1 d ±1, ±ζ 3, ±ζ 3} if d 3 ±ε m d m Z, some unit ε d > 1} if d > 0 see Proositions and of [IR]. In the last case, where Q d is a real number field, the grou of units has infinite order and ε d is called the fundamental unit of Q d. 1 Research artially suorted by AASU Internal Grant #

2 1.. Ramification Theory. Ramification theory was introduced by David Hilbert see [H], [Ja], and [N] and was utilized in his work to find the most general recirocity law in an arbitrary algebraic number field see [T]. For the uroses of this note, we will only consider ramification theory as it ertains to algebraic number fields. Let L/K be an extension of algebraic number fields assume all extensions are Galois, as this is the case we will need and let O L and O K be the resective rings of integers. Then if is a nonzero rime ideal of O K, the ideal O L has the factorization see Theorem 6.8 of [Ja], Chater 1, Section 6 O L P 1 P P g e for distinct rime ideals P i of O L and there is a constant that is indeendent of i. Furthermore, f [O L /P i : O K /] efg [L : K] and the action of GalL/K ermutes the ideals P i of O L containing. Here, e is called the ramification degree of P i over and f is called the residue degree of P i over. Now if P is a nonzero rime ideal of O L and O K P, then we say that P is ramified over O K or ramifies in O L if e > 1. If e 1, then P is called unramified. Examle 1. Consider the uadratic field Qi. In lifting a rime Z to the ring of integers Z[i] Qi, one of three things can occur. It is ossible that the ideal Z[i] is a rime ideal in Z[i]. This is the inert case. It is also ossible that Z[i] where and are distinct rime ideals in Z[i]. Here, we say that slits in Z[i]. Finally, may ramify: Z[i]. The only rime that ramifies in Z[i] is since Z[i] i + 1Z[i]. We leave it as an exercise to rove that slits if 1mod and is inert if 3mod. This result should be comared to the sulementary law to the law of uadratic recirocity the Legendre symbol is defined below: 1 1 Thus, 1 1 if and only if slits in Q 1. 1 if 1mod 1 if 3mod. In general, a similar result holds in Q a for the Legendre symbol: if a Z and is a rational rime such that a, 1, then a 1 if x : amod has a solution 1 otherwise. The following theorem follows from Proosition.1 of [Le]. Theorem. a 1 if slits in Q a 1 if is inert in Q a. In an extension of algebraic number fields L/K, it is known that there are only finitely-many rimes that ramify. This result is obtained by considering the discriminant of a number field and is beyond the scoe of this note. The final result that we will state in this section is due to Kummer cf. Theorem 7., [Ja], Chater 1, Section 7.

3 3 Kummer s Theorem Let be a rime ideal in O K and assume that there is an element θ L such that O L O K [θ]. Let fx be the minimal olynomial of θ over K and let fx be the olynomial obtained by reducing the coefficients of fx modulo. If fx g 1 x a 1 g x a g t x a t is the factorization of fx into a roduct of distinct irreducible olynomials over O K /, then O L P a 1 1 Pa Pa t t for certain rime ideals P i of O L corresonding one-to-one with the irreducible factors g i x. Moreover, the residue degree f of P i corresonds to the degree of g i x Cyclotomic Fields. A cyclotomic field is a field of the form Qζ n we assume n 3, where ζ n cosπ/n + i sinπ/n is a rimitive n th root of unity. We begin with the secial case where n is rime. The minimal olynomial of ζ over Q is Φ x x 1 x 1 x 1 + x + + x + 1, which is irreducible in Z[x] Theorem 1 from [IR], Chater 13, Section. Since the roots of Φ x are the rimitive th roots of unity ζ j where j 1,,..., 1}, we can write 1 3 Φ x x ζ j Qζ [x]. j1 The cyclotomic olynomial Φ n x is defined recursively Proosition 13.. of [IR] by the euation x n 1 d n Φ d x and is the minimal olynomial of ζ n over Q. In general, the ring of integers of an algebraic number field Qα will not be Z[α]. However, in the secial case of Qζ n, we do have that O Qζn Z[ζ n ] Theorem.6 of [W]. Regarding the ramification of rimes in Qζ n over Q, we have the following theorem Proosition.3 of [W]. Theorem 3. ramifies in Qζ n if and only if n. The Galois grou of the extension Qζ n over Q the grou of automorhisms of Qζ n that fix Q is given by GalQζ n /Q σ k k Z/nZ }, where σ k ζ n ζn k [IR], Proosition and the corollaries of Theorem 1 in Chater 13, Section. Under the Fundamental Theorem of Galois Theory, there is is corresondence between subgrous GalQζ n /Q and intermediate fields of the extension [J], Section.5. In the secial case GalQζ /Q, the Galois grou is cyclic and hence has a uniue subgrou of any order that divides GalQζ /Q Z/Z 1.

4 Assuming that 3, we see that 1, and hence, there is a uniue uadratic subfield of Qζ. Plugging x 1 into and 3, we see that 1 ζ1 ζ 1 ζ 1 1 ζ 1 ζ 1 ζ 1 ζ 1 ζ 1 1 ζ 1 ζ ζ ζ ζ ζ 1 ζ 1 1/ 1 1/ ζ j ζ j. j1 Thus, it follows that if 1 1/, then Q is contained in Qζ and if and only if 1mod.. Recirocity Looking back at the develoment of algebraic number theory over the last few centuries, recirocity has influenced the subject more than any other single toic. First observed indeendently by Euler and Legendre, the Law of Quadratic Recirocity demanded a generalization that was sought by number theorists until the 1930s. The Law of Quadratic Recirocity itself has been roved by more than 300 methods and many of the techniues used have rovided the disciline with new tools and at times, comletely new theories. To begin, we state the Law of Quadratic Recirocity. Law of Quadratic Recirocity If and are distinct rational rimes, then The law of uadratic recirocity saw generalizations by individuals such as Eisenstein, Hasse, Hilbert, Takagi, Artin, and Tate. These laws are not the focus of this brief survey and the interested reader is referred to the comrehensive text [Le], as well as its ucoming volumes and 3 art of the draft of volume is currently available on Lemmermeyer s webage htt:// hb3/. The emhasis here will be on a certain class of recirocity laws known as rational recirocity laws..1. Rational Recirocity. The main difference between a recirocity law and a rational recirocity law is that rational recirocity refers to residue symbols that are defined on integers and only take on the values ±1. To begin, we define the rational ower residue symbol, where a, 1, to be 1 if a is an n th ower residue of and 1 otherwise. In articular, if an integer a such that a, 1 satisfies a 1 n 1mod for a rational rime and a nonnegative integer n, then define the rational symbol a a 1 n mod. n This symbol takes on the same value as, the n th ower residue symbol where is any rime a Qζ n above in Qζ n. It should be noted that the Legendre symbol is euivalent to the rational ower residue symbol when n 1. a n

5 5 In 193, Scholz [S] roved a rational uartic recirocity law via class field theory. While the law still bears Scholz s name, it was recently noted by Lemmermeyer see the notes at the end of Chater 5 in [Le] that it had been roved much earlier in 1839 by Schönemann [Sc]. Since then, Scholz s recirocity law has been roved by many different methods see [EP], [L1], [Wi], and [WHF]. The unfamiliar reader is referred to Emma Lehmer s exository article [L] for a comlete descrition of rational recirocity laws and [WHF] or [Le1] for a roof of an all-encomassing rational uartic recirocity law. Scholz s Recirocity Law If 1mod are distinct rimes such that 1, then ε ε. Before we rove Scholz s recirocity law, we begin with a descrition of the uniue uadratic and uartic subfields of Qζ when 1mod. We saw in Section 1.3 that the uadratic subfield is given by K Q. Perhas less well-known, the uartic subfield is given by L Q ε 1 1/ K ε 1 1/ see Proosition 5.9, [Le]. Now we give the roof of Scholz s recirocity law that will be generalized in [BEK]. Proof. We roceed in a manner similar to the roof of uadratic recirocity that was given by Lemmermeyer [Le] after Proosition 3. on age 83. One can easily check that if σ k GalQζ /Q where σ k ζ ζ k, then imlying that Similarly, it can also be shown that σ k GalQζ /K GalQζ /L k, σ k σ k k k } 1. } 1. Thus, the Galois grou GalL/K consists of the identity automorhism and an automorhism α σ k L where k is a uadratic residue of that is not a uartic residue. From these Galois grous, one sees that the cyclotomic olynomial Φ x slits over K as where ϕ x j 1 Φ x ϕ x ϕ x x ζ j and ϕ x k 1 x ζ k. Since Φ x slits into linear factors in Z[ζ ][x], it follows that [ ] 1 + ϕ x, ϕ x Z[ζ ][x] K[x] O K [x] Z [x]. The olynomial ϕ x can then be factored over L: ϕ x ψ x ψ x O L [x]

6 6 where ψ x x ζ m and ψ x x ζ n. m 1 n 1 To simlify notation in what follows, we will denote π ε 1 1/ so that L Q π and hence, L K π and α sends π π. Considering the action of α on the olynomial ϕ x ψ x ψ x, we see that α interchanges ψ x and ψ x. Define the olynomial ϑ x ψ x ψ x and note that Using, it follows that roving that αϑ x ϑ x. α π ϑ x π ϑ x, 5 ϑ x [ ] 1 + π Z [x]. We will write ϑ x π φ x with φ x Z [ 1+ Since 1, slits in K. In other words, we can write λ βλ, where β is the nontrivial automorhism in GalK/Q given by. The residue field O K /λo K Z/Z. Now we raise ] [x]. ϑ x to the ower and reduce modulo λo K. Since we are also assuming that the fact that ±1. 1, we can use 6 ϑ x ψ x ψ x x ζ m m 1 n 1 x ζ n ψ x ψ x mod λo K. The automorhism σ GalQζ /Q is in GalQζ /K and its restriction to GalL/K is α if and only if 1. On the other hand, we can aly 5 to find ϑ x π φ x ε 1 π φ x mod λo K.

7 7 Alying an analog of Fermat s Little Theorem Proosition of [IR] to the coefficients of φ x mod λo K, it follows that 7 ϑ x ε 1 ε ε π φ x ϑ x ψ x ψ x mod λo K. Next, we show that ψ X ψ Xmod λo K. By Kummer s Theorem [Ja], Theorem 7., the ideal generated by in Z[ζ ] which is unramified since is the only ramified rime decomoses in exactly the same way as Φ X decomoses in Z/Z[X]. If ϕ X ψ X mod λo K, then we can ick 0, 1,... 1} as coset reresentatives of O K /λo K Z/Z to obtain a suare factor of Φ X in Z/Z[X], contradicting the observation that does not ramify in Z[ζ ]. Finally, comaring 6 and 7 we obtain ε ε By symmetry, the statement of Scholz s recirocity law follows.... Generalizing Scholz s Law. In [BW1], Buell and Williams conjectured, and in [BW] they roved, an octic recirocity law of Scholz tye which we refer to below as Scholz s octic recirocity law. While this law does not receive as much attention as Scholz s original law, it does rovide insight into the otential formulation of a general rational recirocity law of this tye. Scholz s Octic Recirocity Law such that 1. Then 8 8 Let 1mod 8 and 1mod 8 be distinct rational rimes ε ε 1 h/ ε ε if Nε 1 if Nε 1. Buell and Williams law rovides a beautiful rational octic recirocity law involving the fundamental units of uadratic fields, but it loses some of the simlicity of the statement of Scholz s law. It seems more natural to use units from the uniue uartic subfield of Qζ when constructing such an octic law. This was our motivation in the formulation of the following rational recirocity law similar to that of Scholz ucoming in [BEK]. Theorem. Let 1mod t and be distinct odd rational rimes such that 1, t 1 t 1 t

8 8 and set Then where A t 1 a 1 β t t k } a 1 t 1 t and B t t η t k and η t : β t 1 b 1 t 1 ζ b ζ b b A t 1 B t ζ a ζ a O K. t 1 a A t } b 1. t 1 The roof of Theorem is similar to the roof of Scholz s recirocity law that was given u above. However, one needs to consider the intermediate subfields of degrees t 1 and t K t 1 and K t, resectively of Qζ over Q. It can be shown that η t is a unit in the ring of integers of K t 1, similar to Scholz s law where ε was a unit in the uadratic extension. It is not immediately clear that Theorem contains Scholz s recirocity law as a corollary when t. This realization follows from Proosition 3. of [Le] where it is shown that η ε h for an odd integer h the class number of Q. Hence, we have that η ε h ε, resulting in the statement of Scholz s law. Finally, it should be noted that the octic version t 3 of Theorem is different from that of Buell and Williams and we leave it to the reader to comare the two laws. References [BEK] M. Budden, J. Eisenmenger, and J. Kish, A Generalization of Scholz s Recirocity Law, in rearation [BW1] D. Buell and K. Williams, Is There an Octic Recirocity Law of Scholz Tye?, Amer. Math. Monthly , [BW] D. Buell and K. Williams, An Octic Recirocity Law of Scholz Tye, Proc. Amer. Math. Soc , [EP] D. Estes and G. Pall, Sinor Genera of Binary Quadratic Forms, J. Number Theory , 1-3. [FT] A. Fröhlich and M. Taylor, Algebraic Number Theory, Cambridge University Press, [H] D. Hilbert, The Theory of Algebraic Number Fields, Sringer-Verlag translated by Iain Adamson, Berlin, [IR] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, nd edition, Graduate Texts in Mathematics 8, Sringer-Verlag, New York, [J] N. Jacobson, Basic Algebra I, nd edition, W. H. Freeman and Comany, New York, [Ja] G. Janusz, Algebraic Number Fields, nd ed., Graduate Studies in Mathematics, Vol. 7, American Mathematical Society, Providence, RI, [L1] E. Lehmer, On the Quadratic Character of some Quadratic Surds, J. Reine Angew. Math , -8. [L] E. Lehmer, Rational Recirocity Laws, Amer. Math. Monthly , [Le1] F. Lemmermeyer, Rational Quartic Recirocity, Acta Arith , [Le] F. Lemmermeyer, Recirocity Laws, Sringer Monograhs in Mathematics, Sringer-Verlag, Berlin, 000. [N] J. Neukirch, Algebraic Number Theory, A Series of Comrehensive Studies in Mathematics Vol. 3 translated by Norbert Schaacher, Sringer-Verlag, Berlin, [S] A. Scholz, Über die Lösbarkeit der Gleichung t Du, Math. Z , [Sc] T. Schönemann, Theorie der Symmetrischen Functionen der Wurzeln einer Gleichung. Allgemeine Sätze über Congruenzen nebst einigen Anwendungen derselben, J. Reine Angew. Math , [T] J. Tate, Problem 9: The General Recirocity Law, Mathematical Develoments Arising from Hilbert Problems, Proc. of Sym. in Pure Math , [W] L. Washington, Introduction to Cyclotomic Fields, nd ed., Graduate Texts in Mathematics 83, Sringer-Verlag, New York, 1997.

9 9 [Wi] K. Williams, On Scholz s Recirocity Law, Proc. Amer. Math. Soc. 6 No , 5-6. [WHF] K. Williams, K. Hardy, and C. Friesen, On the Evaluation of the Legendre Symbol 1985, A+B m, Acta Arith. 5 Algebra, Algebraic Number Theory: [FT], [H], [IR], [J], [Ja], [N], [W] Recirocity: [BEK], [BW1], [BW], [EP], [IR], [L1], [L], [Le1], [Le], [S], [Sc], [T], [Wi], [WHF]